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Existence of Hermitian-Yang-Mills metrics under conifold transitions. (English) Zbl 1266.53028
The degeneration of a sequence of Hermitian-Yang-Mills metrics with respect to a sequence of balanced metrics on a Calabi-Yau tree-fold $$X$$ that degenerates to the balanced metric of J. Fu et al. [J. Differ. Geom. 90, No. 1, 81–129 (2012; Zbl 1264.32020)] on the complement of finitely many $$(-1,-1)$$-curves in $$X$$, is studied. Under some assumptions the existence of such metrics is shown over a family of three-folds $$X_t$$ with trivial canonical bundles. These three-folds $$X_t$$ are obtained by performing transitions on $$X$$. It is known [M. Reid, Math. Ann. 278, 329–334 (1987; Zbl 0649.14021)] that the moduli spaces of all Calabi-Yau tree-folds can be connected by means of taking birational transformations and smoothings on the Calabi-Yau tree-folds. This idea, as “Reid’s Fantasy”, was checked for a huge number of examples in [P. Candelas et al., in: Superstrings ’89. Proceedings of the Spring School held in Trieste, Italy, 1989. Singapore: World Scientific. 366–421 (1990; Zbl 0985.32502)]; T.-M. Chiang et al., Nucl. Phys., B, Proc. Suppl. 46, 82–95 (1996; Zbl 0957.32502)].

##### MSC:
 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 81T13 Yang-Mills and other gauge theories in quantum field theory
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